Theory and Examples

In Exerc...

Question

Theory and Examples

In Exercises 51 and 52, give reasons for your answers.

Let f(x) = |x³ − 9x|.

d. Determine all extrema of f.

Accepted Answer

Welcome back, everyone. In this problem, we want to find the local Xtrema of the function F of X equals the absolute value of X 4 minus 8 X. Now for us to find the local extrema of this function, first we'll need to find the critical points from the first derivative, then test the nature of those critical points by using the first derivative test. So let's first start by finding our critical points, and we know that at the critical points of F of X, OK. Then the first derivative of our function is going to be equal to 0. Now how do we differentiate F of X? Well, recall, OK, that if we have a function that's equal to the absolute value of another function GFX. Then it's derivative, OK. The derivative in this case of our function F of X will be equal to the term inside the absolute value G of X divided by the absolute value of GFX multiplied by the derivative of G of X. OK. Now, in this case, that means. If we let G of X be equal to X 4 minus 8 X, then it's derivative. Is going to be equal to 4 x cubed minus 8. Therefore, that means the derivative of our function F of X is going to be equal to G of X X 4 minus 8 X divided by the absolute value of G of X. Multiplied, OK, by the derivative of GFX. In other words, multiplied by 4 X cubed minus 8. Now remember we said that at the critical points this first derivative will be equal to 0. So what will make this be equal to 0? Well, if F of X equals 0, OK. If our derivative equals 0. Then that means either the derivative of G of X will be equal to 0 or G of X is going to be equal to 0. In other words, either both of these terms will be equal to 0, the numerator or for ex cubed minus 8. So let's set both of the of those to 0 and thus solve for the value of X. So if that's the case, that means then that first, if our first derivative of G of X for XQ minus 8 equals 0, then if we divide through by 4 and add 2 to both sides, then X cubed will be equal to 2, which means that X will be equal to the cube root of 2. On the other hand, if G of X equals 0, that means x 4 minus 8 X will be equal to 0, which means that X multiplied by X cubed minus 8 equals 0, and thus that means X will be equal to 0, or X will be equal to the cube root of 8, which is 2. So now we have 3 critical points. This tells us that X equals 0. OK. X equals the cube root of 2, and X equals 2 are all critical points. Now that we have the X values of our critical points, next we need to determine their nature. Uh, we want to figure out if they are maximum or minimum points, and we can do that by using the first derivative test because by that test it basically tells us that if. The left side of our function is of the first derivative is decreasing while the right side is increasing, it's a local minimum. And if the left side is increasing while the right side is decreasing, it's a local maximum. So let's analyze each of those points using that criteria, OK? So now by the first derivative test. OK. Let's see what's happening at X equals 0. OK? Let me, let me separate them here. So let's see what's happening at X equals 0. A X equals the cube root of 2, OK. And Add X equals 2. Let's start with when X equals 0. No, at X equals 0. Let's look, let's take two random points at the left and right sides of uh 0, OK? So, let's take the first derivative when X equals -0.1 for argument's sake. And the for for our left side, OK, so let me just write that. Let me clarify that here. So from the left. Let's use when our first derivative is -0.1 and from the right. OK, on our first derivative, when X is 0.1, OK. Now, what do you think is gonna happen from the left? Well, if we, we can substitute negative 0.1 into our first derivative, but we can also think about it using our science, OK? So we know that this is our first derivative, correct? If we put a negative 0.1 here, OK, then. Our numerator for our quotient is going to be, uh, is going to be positive, OK? Our denominator, we know that's all is going to be positive because it's the absolute value, but our first derivative of geo X is going to be negative because 4 multiplied by a negative value cubed minus another value is going to make our value more negative and positive multiplied by a negative is negative. So we know that when X equals -0.1, OK, then our first derivative is going to be less than 0. OK, so I could actually, let me just even write that here. On the other hand, if X equals 0.1, then our quotient and our term will be positive, which means that it's going to be greater than 0. So that means the left side is decreasing while the right side is increasing. So X equals 0 is a local minimum. And since it's a minimum, that means we can go ahead and find its value by plugging X equals 0 into our function, and no F of 0 will be the absolute value of 0 to 4 minus 8 multiplied by 0, which would be equal to 0. Now we can just repeat the same for the next 2 points. Now at X equals the cube root of 2, let's take 1. For the value to the left. OK. Let's set X to be 1 for the valet to the left, and let's set X to be 1.5 for the value to the, to the right. Now in the interest of time, I'll leave you to try to figure out these signs here or to work it out if you'd like for yourself, but you should find that the left side is increasing, OK, of the first derivative and the right side is decreasing for our first derivative. So thus this means that X equals the cube root of 2 is a local maximum. And now when we go ahead and solve, then F of the cube root of 2 should give you 6 multiplied by 2 to 3 when we when we plug the cube root of 2 into F of X. Finally, at X equals 2, from the left, we could take when X equals 1.9. And from the right, we can take it when X equals 2.1. I know, when X equals 1.9. Our first derivative is negative, so that means the left side is decreasing. While when X equals 2.1, our first derivative is increasing on the right side. So since the left side is decreasing and the right side is increasing, that means X equals 2 corresponds to a local minimum. I know we can find that minimum value by plugging 2 into our function FFX, and we should find that we get FF 2 to be equal to 0. So what can we conclude? Well, no, this tells us then that our local maximum is at, sorry, 6 multiplied by the cube root of 2, at X equals the cube root of 2, and our local minimum is 0, OK, at X equals 0 and X equals 2. Thanks a lot for watching everyone. I hope this video helped.

Theory and Examples

In Exercises 51 and 52, give reasons for your answers.

Let f(x) = |x³ − 9x|.

d. Determine all extrema of f.

Key Concepts

Absolute Value Function

Critical Points

Extrema

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